Nerve complex

<h2 id="basic-definition">Basic definition</h2>
Let 
 
 
 
 I
 
 
 {\displaystyle I}
 
 be a set of indices and 
 
 
 
 C
 
 
 {\displaystyle C}
 
 be a family of sets 
 
 
 
 (
 
 U
 
 i
 
 
 
 )
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle (U_{i})_{i\in I}}
 
. The nerve of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is a set of finite subsets of the index set 
 
 
 
 I
 
 
 {\displaystyle I}
 
. It contains all finite subsets 
 
 
 
 J
 ⊆
 I
 
 
 {\displaystyle J\subseteq I}
 
 such that the intersection of the 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 whose subindices are in 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is non-empty:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>: 81

N
        (
        C
        )
        :=
        
          
            {
          
        
        J
        ⊆
        I
        :
        
          ⋂
          
            j
            ∈
            J
          
        
        
          U
          
            j
          
        
        ≠
        ∅
        ,
        J
        
           finite set
        
        
          
            }
          
        
        .
      
    
    {\displaystyle N(C):={\bigg \{}J\subseteq I:\bigcap _{j\in J}U_{j}\neq \varnothing ,J{\text{ finite set}}{\bigg \}}.}

In Alexandrov's original definition, the sets 
 
 
 
 (
 
 U
 
 i
 
 
 
 )
 
 i
 ∈
 I
 
 
 
 
 {\displaystyle (U_{i})_{i\in I}}
 
 are <a href="/facts/Open_set/QfTCIqMu">open subsets</a> of some topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
.
The set 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 may contain singletons (elements 
 
 
 
 i
 ∈
 I
 
 
 {\displaystyle i\in I}
 
 such that 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 is non-empty), pairs (pairs of elements 
 
 
 
 i
 ,
 j
 ∈
 I
 
 
 {\displaystyle i,j\in I}
 
 such that 
 
 
 
 
 U
 
 i
 
 
 ∩
 
 U
 
 j
 
 
 ≠
 ∅
 
 
 {\displaystyle U_{i}\cap U_{j}\neq \emptyset }
 
), triplets, and so on. If 
 
 
 
 J
 ∈
 N
 (
 C
 )
 
 
 {\displaystyle J\in N(C)}
 
, then any subset of 
 
 
 
 J
 
 
 {\displaystyle J}
 
 is also in 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
, making 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 an <a href="/facts/Abstract_simplicial_complex/ytwxClsf">abstract simplicial complex</a>. Hence N(C) is often called the nerve complex of 
 
 
 
 C
 
 
 {\displaystyle C}
 
.

<h2 id="examples">Examples</h2>
<ol><li>Let X be the circle 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
 and 
 
 
 
 C
 =
 {
 
 U
 
 1
 
 
 ,
 
 U
 
 2
 
 
 }
 
 
 {\displaystyle C=\{U_{1},U_{2}\}}
 
, where 
 
 
 
 
 U
 
 1
 
 
 
 
 {\displaystyle U_{1}}
 
 is an arc covering the upper half of 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
 and 
 
 
 
 
 U
 
 2
 
 
 
 
 {\displaystyle U_{2}}
 
 is an arc covering its lower half, with some overlap at both sides (they must overlap at both sides in order to cover all of 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
). Then 
 
 
 
 N
 (
 C
 )
 =
 {
 {
 1
 }
 ,
 {
 2
 }
 ,
 {
 1
 ,
 2
 }
 }
 
 
 {\displaystyle N(C)=\{\{1\},\{2\},\{1,2\}\}}
 
, which is an abstract 1-simplex.</li>
<li>Let X be the circle 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
 and 
 
 
 
 C
 =
 {
 
 U
 
 1
 
 
 ,
 
 U
 
 2
 
 
 ,
 
 U
 
 3
 
 
 }
 
 
 {\displaystyle C=\{U_{1},U_{2},U_{3}\}}
 
, where each 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
 is an arc covering one third of 
 
 
 
 
 S
 
 1
 
 
 
 
 {\displaystyle S^{1}}
 
, with some overlap with the adjacent 
 
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{i}}
 
. Then 
 
 
 
 N
 (
 C
 )
 =
 {
 {
 1
 }
 ,
 {
 2
 }
 ,
 {
 3
 }
 ,
 {
 1
 ,
 2
 }
 ,
 {
 2
 ,
 3
 }
 ,
 {
 3
 ,
 1
 }
 }
 
 
 {\displaystyle N(C)=\{\{1\},\{2\},\{3\},\{1,2\},\{2,3\},\{3,1\}\}}
 
. Note that {1,2,3} is not in 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 since the common intersection of all three sets is empty; so 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is an unfilled triangle.</li></ol>
<h2 id="the-ech-nerve">The Čech nerve</h2>
Given an <a href="/facts/Open_cover/2VUdelA4">open cover</a> 
 
 
 
 C
 =
 {
 
 U
 
 i
 
 
 :
 i
 ∈
 I
 }
 
 
 {\displaystyle C=\{U_{i}:i\in I\}}
 
 of a topological space 
 
 
 
 X
 
 
 {\displaystyle X}
 
, or more generally a cover in a <a href="/facts/Grothendieck_topology/Przxx5sH">site</a>, we can consider the pairwise <a href="/facts/Pullback_(category_theory)/bC0QoX5U">fibre products</a> 
 
 
 
 
 U
 
 i
 j
 
 
 =
 
 U
 
 i
 
 
 
 ×
 
 X
 
 
 
 U
 
 j
 
 
 
 
 {\displaystyle U_{ij}=U_{i}\times _{X}U_{j}}
 
, which in the case of a topological space are precisely the intersections 
 
 
 
 
 U
 
 i
 
 
 ∩
 
 U
 
 j
 
 
 
 
 {\displaystyle U_{i}\cap U_{j}}
 
. The collection of all such intersections can be referred to as 
 
 
 
 C
 
 ×
 
 X
 
 
 C
 
 
 {\displaystyle C\times _{X}C}
 
 and the triple intersections as 
 
 
 
 C
 
 ×
 
 X
 
 
 C
 
 ×
 
 X
 
 
 C
 
 
 {\displaystyle C\times _{X}C\times _{X}C}
 
.
By considering the natural maps 
 
 
 
 
 U
 
 i
 j
 
 
 →
 
 U
 
 i
 
 
 
 
 {\displaystyle U_{ij}\to U_{i}}
 
 and 
 
 
 
 
 U
 
 i
 
 
 →
 
 U
 
 i
 i
 
 
 
 
 {\displaystyle U_{i}\to U_{ii}}
 
, we can construct a <a href="/facts/Simplicial_object/6DaqXMLT">simplicial object</a> 
 
 
 
 S
 (
 C
 
 )
 
 ∙
 
 
 
 
 {\displaystyle S(C)_{\bullet }}
 
 defined by 
 
 
 
 S
 (
 C
 
 )
 
 n
 
 
 =
 C
 
 ×
 
 X
 
 
 ⋯
 
 ×
 
 X
 
 
 C
 
 
 {\displaystyle S(C)_{n}=C\times _{X}\cdots \times _{X}C}
 
, n-fold fibre product. This is the Čech nerve.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> 
By taking connected components we get a <a href="/facts/Simplicial_set/6DaqXMLT">simplicial set</a>, which we can realise topologically: 
 
 
 
 
 |
 
 S
 (
 
 π
 
 0
 
 
 (
 C
 )
 )
 
 |
 
 
 
 {\displaystyle |S(\pi _{0}(C))|}
 
.

<h2 id="nerve-theorems">Nerve theorems</h2>
The nerve complex 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is a simple combinatorial object. Often, it is much simpler than the underlying topological space (the union of the sets in 
 
 
 
 C
 
 
 {\displaystyle C}
 
). Therefore, a natural question is whether the topology of 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is equivalent to the topology of 
 
 
 
 ⋃
 C
 
 
 {\displaystyle \bigcup C}
 
. 
In general, this need not be the case. For example, one can cover any <a href="/facts/N-sphere/bFu2S7E2">n-sphere</a> with two contractible sets 
 
 
 
 
 U
 
 1
 
 
 
 
 {\displaystyle U_{1}}
 
 and 
 
 
 
 
 U
 
 2
 
 
 
 
 {\displaystyle U_{2}}
 
 that have a non-empty intersection, as in example 1 above. In this case, 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is an abstract 1-simplex, which is similar to a line but not to a sphere.
However, in some cases 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 does reflect the topology of X. For example, if a circle is covered by three open arcs, intersecting in pairs as in Example 2 above, then 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is a 2-simplex (without its interior) and it is <a href="/facts/Homotopy-equivalent/1nWrPxdj">homotopy-equivalent</a> to the original circle.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
A nerve theorem (or nerve lemma) is a theorem that gives sufficient conditions on C guaranteeing that 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 reflects, in some sense, the topology of 
 
 
 
 ⋃
 C
 
 
 {\displaystyle \bigcup C}
 
. A functorial nerve theorem is a nerve theorem that is functorial in an appropriate sense, which is, for example, crucial in <a href="/facts/Topological_data_analysis/Lz5ltT7l">topological data analysis</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<h3>Leray's nerve theorem</h3>
The basic nerve theorem of <a href="/facts/Jean_Leray/MyAEDyc7">Jean Leray</a> says that, if any intersection of sets in 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is <a href="/facts/Contractible_space/REXYj8uS">contractible</a> (equivalently: for each finite 
 
 
 
 J
 ⊂
 I
 
 
 {\displaystyle J\subset I}
 
 the set 
 
 
 
 
 ⋂
 
 i
 ∈
 J
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle \bigcap _{i\in J}U_{i}}
 
 is either empty or contractible; equivalently: C is a <a href="/facts/Good_cover/Km2ckrLD">good open cover</a>), then 
 
 
 
 N
 (
 C
 )
 
 
 {\displaystyle N(C)}
 
 is <a href="/facts/Homotopy-equivalent/1nWrPxdj">homotopy-equivalent</a> to 
 
 
 
 ⋃
 C
 
 
 {\displaystyle \bigcup C}
 
.

<h3>Borsuk's nerve theorem</h3>
There is a discrete version, which is attributed to <a href="/facts/Karol_Borsuk/SNqCQ4EB">Borsuk</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a><a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>: 81, Thm.4.4.4  Let K1,...,Kn be <a href="/facts/Abstract_simplicial_complex/ytwxClsf">abstract simplicial complexes</a>, and denote their union by K. Let Ui = ||Ki|| = the <a href="/facts/Abstract_simplicial_complex/ytwxClsf">geometric realization</a> of Ki, and denote the nerve of {U1, ... , Un } by N.
If, for each nonempty 
 
 
 
 J
 ⊂
 I
 
 
 {\displaystyle J\subset I}
 
, the intersection 
 
 
 
 
 ⋂
 
 i
 ∈
 J
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle \bigcap _{i\in J}U_{i}}
 
 is either empty or contractible, then N is <a href="/facts/Homotopy-equivalent/1nWrPxdj">homotopy-equivalent</a> to K.
A stronger theorem was proved by <a href="/facts/Anders_Bj%25C3%25B6rner/Po522H5D">Anders Bjorner</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> if, for each nonempty 
 
 
 
 J
 ⊂
 I
 
 
 {\displaystyle J\subset I}
 
, the intersection 
 
 
 
 
 ⋂
 
 i
 ∈
 J
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle \bigcap _{i\in J}U_{i}}
 
 is either empty or <a href="/facts/N-connected_space/dbCgWgu3">(k-|J|+1)-connected</a>, then for every j ≤ k, the j-th <a href="/facts/Homotopy_group/xjogL95Y">homotopy group</a> of N is isomorphic to the j-th <a href="/facts/Homotopy_group/xjogL95Y">homotopy group</a> of K. In particular, N is k-connected if-and-only-if K is k-connected.

<h3>Čech nerve theorem</h3>
Another nerve theorem relates to the Čech nerve above: if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is compact and all intersections of sets in C are contractible or empty, then the space 
 
 
 
 
 |
 
 S
 (
 
 π
 
 0
 
 
 (
 C
 )
 )
 
 |
 
 
 
 {\displaystyle |S(\pi _{0}(C))|}
 
 is <a href="/facts/Homotopy-equivalent/1nWrPxdj">homotopy-equivalent</a> to 
 
 
 
 X
 
 
 {\displaystyle X}
 
.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

<h3>Homological nerve theorem</h3>
The following nerve theorem uses the <a href="/facts/Homology_groups/VFJmryEW">homology groups</a> of intersections of sets in the cover.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> For each finite 
 
 
 
 J
 ⊂
 I
 
 
 {\displaystyle J\subset I}
 
, denote 
 
 
 
 
 H
 
 J
 ,
 j
 
 
 :=
 
 
 
 
 H
 ~
 
 
 
 
 j
 
 
 (
 
 ⋂
 
 i
 ∈
 J
 
 
 
 U
 
 i
 
 
 )
 =
 
 
 {\displaystyle H_{J,j}:={\tilde {H}}_{j}(\bigcap _{i\in J}U_{i})=}
 
 the j-th <a href="/facts/Reduced_homology/ySoPPB63">reduced homology</a> group of 
 
 
 
 
 ⋂
 
 i
 ∈
 J
 
 
 
 U
 
 i
 
 
 
 
 {\displaystyle \bigcap _{i\in J}U_{i}}
 
.
If HJ,j is the <a href="/facts/Trivial_group/XPjQkQFm">trivial group</a> for all J in the k-skeleton of N(C) and for all j in {0, ..., k-dim(J)}, then N(C) is "homology-equivalent" to X in the following sense:

<ul><li>
 
 
 
 
 
 
 
 H
 ~
 
 
 
 
 j
 
 
 (
 N
 (
 C
 )
 )
 ≅
 
 
 
 
 H
 ~
 
 
 
 
 j
 
 
 (
 X
 )
 
 
 {\displaystyle {\tilde {H}}_{j}(N(C))\cong {\tilde {H}}_{j}(X)}
 
 for all j in {0, ..., k};</li>
<li>if 
 
 
 
 
 
 
 
 H
 ~
 
 
 
 
 k
 +
 1
 
 
 (
 N
 (
 C
 )
 )
 ≇
 0
 
 
 {\displaystyle {\tilde {H}}_{k+1}(N(C))\not \cong 0}
 
 then 
 
 
 
 
 
 
 
 H
 ~
 
 
 
 
 k
 +
 1
 
 
 (
 X
 )
 ≇
 0
 
 
 {\displaystyle {\tilde {H}}_{k+1}(X)\not \cong 0}
 
 .</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Aleksandroff, P. S. (1928). "Über den allgemeinen Dimensionsbegriff und seine Beziehungen zur elementaren geometrischen Anschauung". Mathematische Annalen. 98: 617–635. doi:10.1007/BF01451612. S2CID 119590045. <a href="/wiki/Pavel_Alexandrov" target="_blank">/wiki/Pavel_Alexandrov</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Eilenberg, Samuel; Steenrod, Norman (1952-12-31). Foundations of Algebraic Topology. Princeton: Princeton University Press. doi:10.1515/9781400877492. ISBN 978-1-4008-7749-2. {{cite book}}: ISBN / Date incompatibility (help) <a href="978-1-4008-7749-2" target="_blank">978-1-4008-7749-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler
, Section 4.3 <a href="978-3-540-00362-5" target="_blank">978-3-540-00362-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Čech nerve in nLab". ncatlab.org. Retrieved 2020-08-07. <a href="https://ncatlab.org/nlab/show/%C4%8Cech+nerve" target="_blank">https://ncatlab.org/nlab/show/%C4%8Cech+nerve</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Artin, Michael; Mazur, Barry (1969). Etale Homotopy. Lecture Notes in Mathematics. Vol. 100. doi:10.1007/bfb0080957. ISBN 978-3-540-04619-6. ISSN 0075-8434. <a href="978-3-540-04619-6" target="_blank">978-3-540-04619-6</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bauer, Ulrich; Kerber, Michael; Roll, Fabian; Rolle, Alexander (2023). "A unified view on the functorial nerve theorem and its variations". Expositiones Mathematicae. 41 (4). arXiv:2203.03571. doi:10.1016/j.exmath.2023.04.005. <a href="/wiki/Expositiones_Mathematicae" target="_blank">/wiki/Expositiones_Mathematicae</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Borsuk, Karol (1948). "On the imbedding of systems of compacta in simplicial complexes". Fundamenta Mathematicae. 35 (1): 217–234. doi:10.4064/fm-35-1-217-234. ISSN 0016-2736. <a href="https://eudml.org/doc/213158" target="_blank">https://eudml.org/doc/213158</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler
, Section 4.3 <a href="978-3-540-00362-5" target="_blank">978-3-540-00362-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Björner, Anders (2003-04-01). "Nerves, fibers and homotopy groups". Journal of Combinatorial Theory. Series A. 102 (1): 88–93. doi:10.1016/S0097-3165(03)00015-3. ISSN 0097-3165. <a href="/wiki/Anders_Bj%C3%B6rner" target="_blank">/wiki/Anders_Bj%C3%B6rner</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Nerve theorem at the nLab <a href="https://ncatlab.org/nlab/show/nerve+theorem" target="_blank">https://ncatlab.org/nlab/show/nerve+theorem</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912. S2CID 207006642. <a href="/wiki/Combinatorica" target="_blank">/wiki/Combinatorica</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Nerve complex open-in-new

Nerve complex